Russ,
Very nice calculations. It would have taken me quite a while to
figure it out. Thanks!
Tom,
Interesting article. At the end though, I
think the review's author misses the point of why Bayes theorem was so
controversial amongst the 'frequents' (which suggests the book's author might
have missed the point as well). The controversy occurred because many early
statisticians wanted to believe in a truly probabilistic future, but believed
in an already determined past -- basically what most people believe in. In that
sense, there is a probability that you will pick the counterfeit coin before
you make the choice, the 'a priori' probability (given a random choice) is .33.
After you pick a coin, either you picked the conterfeit one or you did not,
thus there is no probability worth discussing; the 'a posteriori' probability
that you picked the counterfeit coin is either 1 or 0. Once the coin is in my
hand, and no matter how many times I flip it, there will never be a 4/5ths
chance that I picked the counterfeit coin. What would that even mean?!? Or so
the anti-Bayes people argued: We can talk about our best guess as to the truth
all day, but we are NOT talking about probability when we do so.
Fisher tried to deal with
the problem of using the present to guess the past with his 'likelihood'
formulae. Under some circumstances likelihood and Bayes theorem will come to
the same number, but other times they will not. Likelihood calculations are
still around, but not as popular as Bayes, because it is much harder to derive
the formulae (and sometimes harder to gather the needed data). Fun fact:
Researcher's reliance on Bayes formula was what lead Fisher to insist
throughout his life that there was no evidence that smoking caused cancer.
There is now evidence he would accept, but no data at the time allowed what he
deemed to be the proper calculations.
Eric
P.S. For any stats
people who might be reading, I have published on the problem of creating
confidence intervals around correlations corrected for attenuation due to
measurement error. If your population correlation is near 0, then the
probability distribution for sample correlations is symmetric, and likelihood
and Bayes will give you the same answer. As you approach 1 (or -1), the
probability distributions becomes highly asymmetric, and likelihood and Bayes
will give quite different answers. (Confession of mathematical inadequacies: I
tackled the problem through simulation, not through
derivation).
On Sun, Aug 7, 2011 07:48 PM,
Russ Abbott
<[hidden email]> wrote:
When
I read that review it wasn't obvious to me how he got the result that he did
for the counterfeit coin example. So I worked it out for
myself--and after a bit of thinking about it got the same answer. If
you're interested it's <a href="http://cs.calstatela.edu/wiki/index.php/Bayes%27_theorem#Counterfeit_coin_example" onclick="window.open('http://cs.calstatela.edu/wiki/index.php/Bayes%27_theorem#Counterfeit_coin_example');return false;">here. (Let me know if you think I made any mistakes.) The calculation is at the bottom of the Bayes Theorem page on my wiki.
-- Russ Abbott
_____________________________________________ Professor, Computer Science
California State University, Los Angeles
Google voice: 747-999-5105
blog: <a href="http://russabbott.blogspot.com/" style="font-style: italic;" target="" onclick="window.open('http://russabbott.blogspot.com/');return false;">http://russabbott.blogspot.com/
vita: <a href="http://sites.google.com/site/russabbott/" style="font-style: italic;" target="" onclick="window.open('http://sites.google.com/site/russabbott/');return false;">http://sites.google.com/site/russabbott/
_____________________________________________
On Sun, Aug 7, 2011 at 2:41 PM, Tom Johnson
<<a href="x-msg://10/#">tom@...> wrote:
A review of a new book that may be of interest.
--tom johnson
The Mathematics of Changing Your Mind
By JOHN ALLEN PAULOS
Published: August 5, 2011
Sharon Bertsch McGrayne introduces Bayes’s theorem in her new book with a remark by John Maynard Keynes: “When the facts change, I change my opinion. What do you do, sir?”
Bayes’s theorem, named after the 18th-century Presbyterian minister Thomas Bayes, addresses this selfsame essential task: How should we modify our beliefs in the light of additional information? Do we cling to old assumptions long after they’ve become untenable, or abandon them too readily at the first whisper of doubt? Bayesian reasoning promises to bring our views gradually into line with reality and so has become an invaluable tool for scientists of all sorts and, indeed, for anyone who wants, putting it grandiloquently, to sync up with the universe. If you are not thinking like a Bayesian, perhaps you should be.
<a href="http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&ref=books" target="" onclick="window.open('http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&ref=books');return false;">http://www.nytimes.com/2011/08/07/books/review/the-theory-that-would-not-die-by-sharon-bertsch-mcgrayne-book-review.html?_r=1&ref=books
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Eric Charles
Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601